TPTP Problem File: ITP059^2.p
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%------------------------------------------------------------------------------
% File : ITP059^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer FLPTheorem problem prob_747__3300826_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : FLPTheorem/prob_747__3300826_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 318 ( 120 unt; 50 typ; 0 def)
% Number of atoms : 601 ( 203 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3600 ( 85 ~; 15 |; 42 &;3150 @)
% ( 0 <=>; 308 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 156 ( 156 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 8 con; 0-7 aty)
% Number of variables : 638 ( 17 ^; 549 !; 35 ?; 638 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:46.464
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext,type,
configuration_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_AsynchronousSystem_Omessage,type,
message: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_v,type,
v: $tType ).
thf(ty_tf_s,type,
s: $tType ).
thf(ty_tf_p,type,
p: $tType ).
% Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_c_AsynchronousSystem_Oconfiguration_Omsgs,type,
msgs:
!>[P: $tType,V: $tType,S: $tType,Z: $tType] : ( ( configuration_ext @ P @ V @ S @ Z ) > ( message @ P @ V ) > nat ) ).
thf(sy_c_AsynchronousSystem_Oenabled,type,
enabled:
!>[P: $tType,V: $tType,S: $tType] : ( ( configuration_ext @ P @ V @ S @ product_unit ) > ( message @ P @ V ) > $o ) ).
thf(sy_c_AsynchronousSystem_OisReceiverOf,type,
isReceiverOf:
!>[P: $tType,V: $tType] : ( P > ( message @ P @ V ) > $o ) ).
thf(sy_c_Execution_Oexecution_OfirstOccurrence,type,
firstOccurrence:
!>[P: $tType,V: $tType,S: $tType] : ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( message @ P @ V ) > nat > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_ListUtilities_OprefixList,type,
prefixList:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_fe____,type,
fe: nat > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
thf(sy_v_firstOccSet____,type,
firstOccSet: nat > ( set @ ( message @ p @ v ) ) ).
thf(sy_v_ft____,type,
ft: nat > ( list @ ( message @ p @ v ) ) ).
thf(sy_v_index____,type,
index: nat ).
thf(sy_v_msgInSet____,type,
msgInSet: message @ p @ v ).
thf(sy_v_msg____,type,
msg: message @ p @ v ).
thf(sy_v_n0____,type,
n0: nat ).
thf(sy_v_n1____,type,
n1: nat ).
thf(sy_v_nMsg____,type,
nMsg: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p2: p ).
% Relevant facts (251)
thf(fact_0_AssumpOcc6_I1_J,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ nMsg ).
% AssumpOcc6(1)
thf(fact_1_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I2_J,axiom,
! [P2: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,Q: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,A4: configuration_ext @ p @ v @ s @ product_unit,A5: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),A6: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),A7: nat] :
( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( P2 @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( P2 @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( Q @ A4 @ A5 @ A6 @ A7 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(2)
thf(fact_2_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I1_J,axiom,
! [P2: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,Q: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,A0: configuration_ext @ p @ v @ s @ product_unit,A1: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),A2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),A3: nat] :
( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( P2 @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( P2 @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
=> ( ! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( P2 @ A0 @ A1 @ A2 @ A3 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(1)
thf(fact_3_AssumptionSubset_I3_J,axiom,
firstOccurrence @ p @ v @ s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msg @ nMsg ).
% AssumptionSubset(3)
thf(fact_4_SmallIndex,axiom,
! [NMsg: nat] :
( ( firstOccurrence @ p @ v @ s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msg @ NMsg )
=> ( ord_less @ nat @ NMsg @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) ) ) ).
% SmallIndex
thf(fact_5_AssumptionSubset3_I3_J,axiom,
ord_less @ nat @ nMsg @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) ).
% AssumptionSubset3(3)
thf(fact_6_AssumptionSubset2_I3_J,axiom,
ord_less @ nat @ n1 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) ).
% AssumptionSubset2(3)
thf(fact_7_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_8_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_9_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_10_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_11_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A8 @ B ) )
= ( ord_less @ A @ B @ A8 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_12_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_13_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_14_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_15_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_16_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_17_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_18_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_19_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_20_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_21_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ A8 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_22_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ ( zero_zero @ A ) )
= A8 ) ) ).
% diff_zero
thf(fact_23_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A8 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_24_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ ( zero_zero @ A ) )
= A8 ) ) ).
% diff_0_right
thf(fact_25_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ A8 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_26_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_27_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_28_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_29_bot__nat__0_Onot__eq__extremum,axiom,
! [A8: nat] :
( ( A8
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A8 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_30_AssumptionSubset_I2_J,axiom,
firstOccurrence @ p @ v @ s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msgInSet @ n1 ).
% AssumptionSubset(2)
thf(fact_31_AssumptionSubset_I1_J,axiom,
ord_less_eq @ nat @ n1 @ nMsg ).
% AssumptionSubset(1)
thf(fact_32_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_33_measure__induct__rule,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F: A > B2,P2: A > $o,A8: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P2 @ Y ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A8 ) ) ) ).
% measure_induct_rule
thf(fact_34_measure__induct,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F: A > B2,P2: A > $o,A8: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P2 @ Y ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A8 ) ) ) ).
% measure_induct
thf(fact_35_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A8: A,C: A,B: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A8 @ C ) @ B )
= ( minus_minus @ A @ ( minus_minus @ A @ A8 @ B ) @ C ) ) ) ).
% diff_right_commute
thf(fact_36_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A,B: A,C: A,D: A] :
( ( ( minus_minus @ A @ A8 @ B )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A8 = B )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_37_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_38_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_39_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V2: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P2 @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V2 @ Y ) @ ( V2 @ X3 ) )
& ~ ( P2 @ Y ) ) )
=> ( P2 @ X ) ) ).
% infinite_descent_measure
thf(fact_40_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ord_less @ nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_41_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_42_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_43_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_44_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less @ nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_45_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_46_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_47_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less @ nat @ M @ N2 )
| ( ord_less @ nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_48_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y3: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y3 ) )
=> ( X != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_49_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_50_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_51_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ M @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_52_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_53_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_54_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A9: A,B3: A] :
( ( minus_minus @ A @ A9 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_55_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,C: A] :
( ( ord_less @ A @ A8 @ B )
=> ( ord_less @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_56_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A8: A,C: A] :
( ( ord_less @ A @ B @ A8 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A8 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% diff_strict_left_mono
thf(fact_57_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,C: A,D: A] :
( ( ( minus_minus @ A @ A8 @ B )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A8 @ B )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_58_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,D: A,C: A] :
( ( ord_less @ A @ A8 @ B )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_59_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_60_old_Onat_Oinducts,axiom,
! [P2: nat > $o,Nat: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P2 @ Nat3 )
=> ( P2 @ ( suc @ Nat3 ) ) )
=> ( P2 @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_61_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_62_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_63_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_64_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_65_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_66_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N2: nat] :
( ! [X3: nat] : ( P2 @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y5: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] :
( ( P2 @ X3 @ Y5 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
=> ( P2 @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_67_nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) )
=> ( P2 @ N2 ) ) ) ).
% nat_induct
thf(fact_68_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_69_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_70_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_71_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_72_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_73_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_74_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_75_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_76_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_77_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_78_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_79_All__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_80_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_81_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_82_Ex__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N2 )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_83_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_84_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_85_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_86_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_87_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_88_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_89_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P2: A > $o,X: A] :
( ! [X3: A] :
( ( ( V2 @ X3 )
= ( zero_zero @ nat ) )
=> ( P2 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
=> ( ~ ( P2 @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V2 @ Y ) @ ( V2 @ X3 ) )
& ~ ( P2 @ Y ) ) ) )
=> ( P2 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_90_bot__nat__0_Oextremum__strict,axiom,
! [A8: nat] :
~ ( ord_less @ nat @ A8 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_91_infinite__descent0,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( P2 @ N )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_92_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_93_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_94_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_95_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_96_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_97_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_98_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_99_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M )
= ( zero_zero @ nat ) )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_100_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_101_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_102_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A9: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A9 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_103_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_104_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less @ nat @ N2 @ N3 )
=> ( ord_less @ A @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_105_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_106_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_107_All__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_108_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_109_Ex__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_110_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_111_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_112_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_113_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_114_SameCfgOnLow,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) )
=> ( ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) @ I4 )
= ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ I4 ) ) ) ).
% SameCfgOnLow
thf(fact_115__092_060open_062fe_Aindex_A_B_AnMsg_A_061_Afe_A_ISuc_Aindex_J_A_B_AnMsg_092_060close_062,axiom,
( ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) @ nMsg )
= ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ nMsg ) ) ).
% \<open>fe index ! nMsg = fe (Suc index) ! nMsg\<close>
thf(fact_116_ShorterTrace,axiom,
ord_less @ nat @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ ( suc @ index ) ) ) ).
% ShorterTrace
thf(fact_117_NotEmpty_I1_J,axiom,
( ( fe @ ( suc @ index ) )
!= ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ).
% NotEmpty(1)
thf(fact_118_IPrefixList,axiom,
! [I4: nat] : ( prefixList @ ( message @ p @ v ) @ ( ft @ I4 ) @ ( ft @ ( suc @ I4 ) ) ) ).
% IPrefixList
thf(fact_119_IPrefixListEx,axiom,
! [I4: nat] : ( prefixList @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ I4 ) @ ( fe @ ( suc @ I4 ) ) ) ).
% IPrefixListEx
thf(fact_120_NatPredicateTippingPoint,axiom,
! [N22: nat,Pr: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N22 )
=> ( ( Pr @ ( zero_zero @ nat ) )
=> ( ~ ( Pr @ N22 )
=> ? [N: nat] :
( ( ord_less @ nat @ N @ N22 )
& ( Pr @ N )
& ~ ( Pr @ ( suc @ N ) ) ) ) ) ) ).
% NatPredicateTippingPoint
thf(fact_121_NotEmpty_I2_J,axiom,
( ( fe @ index )
!= ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ).
% NotEmpty(2)
thf(fact_122_AssumptionFair_I4_J,axiom,
isReceiverOf @ p @ v @ p2 @ msg ).
% AssumptionFair(4)
thf(fact_123_AssumpOcc6_I2_J,axiom,
( msg
!= ( nth @ ( message @ p @ v ) @ ( ft @ index ) @ ( minus_minus @ nat @ nMsg @ ( one_one @ nat ) ) ) ) ).
% AssumpOcc6(2)
thf(fact_124_Occ1,axiom,
? [P3: p] : ( isReceiverOf @ p @ v @ P3 @ msg ) ).
% Occ1
thf(fact_125_AssumptionSubset2_I1_J,axiom,
? [P3: p] : ( isReceiverOf @ p @ v @ P3 @ msgInSet ) ).
% AssumptionSubset2(1)
thf(fact_126_SameMsgOnLow,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) )
=> ( ( nth @ ( message @ p @ v ) @ ( ft @ index ) @ I4 )
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% SameMsgOnLow
thf(fact_127_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_128_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_129_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_130_bot__nat__0_Oextremum,axiom,
! [A8: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A8 ) ).
% bot_nat_0.extremum
thf(fact_131_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_132__092_060open_062length_A_Ife_Aindex_J_A_N_A1_A_092_060le_062_Alength_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_092_060close_062,axiom,
ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) @ ( one_one @ nat ) ) ).
% \<open>length (fe index) - 1 \<le> length (fe (Suc index)) - 1\<close>
thf(fact_133__092_060open_062_092_060not_062_A_I_092_060exists_062i_060length_A_Ift_A_ISuc_Aindex_J_J_O_Alength_A_Ift_Aindex_J_A_092_060le_062_Ai_A_092_060and_062_Amsg_A_061_Aft_A_ISuc_Aindex_J_A_B_Ai_J_092_060close_062,axiom,
~ ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ ( suc @ index ) ) ) )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) @ I4 )
& ( msg
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% \<open>\<not> (\<exists>i<length (ft (Suc index)). length (ft index) \<le> i \<and> msg = ft (Suc index) ! i)\<close>
thf(fact_134_NotConsumedIntermediate,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ ( suc @ index ) ) ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) @ I4 )
=> ( ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ I4 )
!= msg ) ) ) ).
% NotConsumedIntermediate
thf(fact_135_OccSameMsg,axiom,
! [N4: nat] :
( ( ord_less_eq @ nat @ nMsg @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) )
=> ( ( nth @ ( message @ p @ v ) @ ( ft @ index ) @ N4 )
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% OccSameMsg
thf(fact_136_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A8 @ B ) )
= ( ord_less_eq @ A @ B @ A8 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_137_Occ5,axiom,
! [N4: nat] :
( ( ord_less_eq @ nat @ nMsg @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ index ) ) )
=> ( msg
!= ( nth @ ( message @ p @ v ) @ ( ft @ index ) @ N4 ) ) ) ) ).
% Occ5
thf(fact_138_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_139_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_140_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_141_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
= N2 ) ).
% diff_Suc_1
thf(fact_142_AssumptionSubset3_I5_J,axiom,
! [N4: nat] :
( ( ord_less_eq @ nat @ nMsg @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ ( suc @ index ) ) ) )
=> ( msg
!= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% AssumptionSubset3(5)
thf(fact_143_AssumptionSubset2_I5_J,axiom,
! [N4: nat] :
( ( ord_less_eq @ nat @ n1 @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ ( suc @ index ) ) ) )
=> ( msgInSet
!= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% AssumptionSubset2(5)
thf(fact_144__092_060open_062_092_060not_062_A_I_092_060exists_062i_060length_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_O_Alength_A_Ife_Aindex_J_A_N_A1_A_092_060le_062_Ai_A_092_060and_062_Amsg_A_061_Aft_A_ISuc_Aindex_J_A_B_Ai_J_092_060close_062,axiom,
~ ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) @ ( one_one @ nat ) ) )
& ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) @ ( one_one @ nat ) ) @ I4 )
& ( msg
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% \<open>\<not> (\<exists>i<length (fe (Suc index)) - 1. length (fe index) - 1 \<le> i \<and> msg = ft (Suc index) ! i)\<close>
thf(fact_145_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_146_AssumptionFairContr,axiom,
! [N4: nat] :
( ( ord_less_eq @ nat @ n @ N4 )
=> ! [N0: nat] :
( ( ord_less @ nat @ N0 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ N4 ) ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ n ) ) @ N0 )
=> ( msg
!= ( nth @ ( message @ p @ v ) @ ( ft @ N4 ) @ N0 ) ) ) ) ) ).
% AssumptionFairContr
thf(fact_147_AssumpOcc6_I3_J,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) @ ( minus_minus @ nat @ nMsg @ ( one_one @ nat ) ) ) @ msg ).
% AssumpOcc6(3)
thf(fact_148_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_149_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_150_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_151_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_152_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
| ( ord_less_eq @ nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_153_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_154_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_155_PrefixListTransitive,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,L3: list @ A] :
( ( prefixList @ A @ L1 @ L2 )
=> ( ( prefixList @ A @ L2 @ L3 )
=> ( prefixList @ A @ L1 @ L3 ) ) ) ).
% PrefixListTransitive
thf(fact_156_MinPredicate,axiom,
! [P2: nat > $o] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N02: nat] :
( ( P2 @ N02 )
& ! [N4: nat] :
( ( P2 @ N4 )
=> ( ord_less_eq @ nat @ N02 @ N4 ) ) ) ) ).
% MinPredicate
thf(fact_157_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq @ nat @ N2 @ N3 )
=> ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_158_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ N3 )
=> ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_159_KeepProperty,axiom,
! [Low: nat,P2: nat > $o,Q: nat > $o] :
( ! [I2: nat] :
( ( ord_less_eq @ nat @ Low @ I2 )
=> ( ( P2 @ I2 )
=> ( ( P2 @ ( suc @ I2 ) )
& ( Q @ I2 ) ) ) )
=> ( ( P2 @ Low )
=> ! [I4: nat] :
( ( ord_less_eq @ nat @ Low @ I4 )
=> ( Q @ I4 ) ) ) ) ).
% KeepProperty
thf(fact_160_PrefixSameOnLow,axiom,
! [A: $tType,L1: list @ A,L2: list @ A] :
( ( prefixList @ A @ L1 @ L2 )
=> ! [Index: nat] :
( ( ord_less @ nat @ Index @ ( size_size @ ( list @ A ) @ L1 ) )
=> ( ( nth @ A @ L1 @ Index )
= ( nth @ A @ L2 @ Index ) ) ) ) ).
% PrefixSameOnLow
thf(fact_161_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_162_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,D: A,C: A] :
( ( ord_less_eq @ A @ A8 @ B )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% diff_mono
thf(fact_163_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A8: A,C: A] :
( ( ord_less_eq @ A @ B @ A8 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A8 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% diff_left_mono
thf(fact_164_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,C: A] :
( ( ord_less_eq @ A @ A8 @ B )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% diff_right_mono
thf(fact_165_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B: A,C: A,D: A] :
( ( ( minus_minus @ A @ A8 @ B )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A8 @ B )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_166_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_167_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_168_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_169_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_170_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq @ nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_171_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_172_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N2 ) )
= ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_173_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_174_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P2 @ M )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_175_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( R @ X3 @ Y5 )
=> ( ( R @ Y5 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_176_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_177_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_178_bot__nat__0_Oextremum__unique,axiom,
! [A8: nat] :
( ( ord_less_eq @ nat @ A8 @ ( zero_zero @ nat ) )
= ( A8
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_179_bot__nat__0_Oextremum__uniqueI,axiom,
! [A8: nat] :
( ( ord_less_eq @ nat @ A8 @ ( zero_zero @ nat ) )
=> ( A8
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_180_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_181_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_182_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_183_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_184_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_185_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_186_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_187_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_188_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_189_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_190_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_191_le__diff__iff_H,axiom,
! [A8: nat,C: nat,B: nat] :
( ( ord_less_eq @ nat @ A8 @ C )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A8 ) @ ( minus_minus @ nat @ C @ B ) )
= ( ord_less_eq @ nat @ B @ A8 ) ) ) ) ).
% le_diff_iff'
thf(fact_192_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_193_MinPredicate2,axiom,
! [P2: nat > $o] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N02: nat] :
( ( P2 @ N02 )
& ( ( N02
= ( zero_zero @ nat ) )
| ~ ( P2 @ ( minus_minus @ nat @ N02 @ ( one_one @ nat ) ) ) ) ) ) ).
% MinPredicate2
thf(fact_194_PrefixListMonotonicity,axiom,
! [A: $tType,L1: list @ A,L2: list @ A] :
( ( prefixList @ A @ L1 @ L2 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ L1 ) @ ( size_size @ ( list @ A ) @ L2 ) ) ) ).
% PrefixListMonotonicity
thf(fact_195_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A9: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A9 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_196_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_197_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_198_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( ord_less @ nat @ N @ J )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_199_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( ord_less @ nat @ N @ J )
=> ( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_200_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_201_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_202_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_203_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N5: nat] : ( ord_less_eq @ nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_204_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_205_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_206_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_207_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_208_diff__less__mono,axiom,
! [A8: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A8 @ B )
=> ( ( ord_less_eq @ nat @ C @ A8 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A8 @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_209_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_210_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_211_ex__least__nat__less,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_212_nat__induct__non__zero,axiom,
! [N2: nat,P2: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_213_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( N2
= ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_214_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_215_AssumptionSubset2_I6_J,axiom,
( ( n1
!= ( zero_zero @ nat ) )
=> ( ~ ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ ( minus_minus @ nat @ n1 @ ( one_one @ nat ) ) ) @ msgInSet )
| ( msgInSet
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ ( minus_minus @ nat @ n1 @ ( one_one @ nat ) ) ) ) ) ) ).
% AssumptionSubset2(6)
thf(fact_216_AssumptionSubset3_I6_J,axiom,
( ( nMsg
!= ( zero_zero @ nat ) )
=> ( ~ ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ ( minus_minus @ nat @ nMsg @ ( one_one @ nat ) ) ) @ msg )
| ( msg
= ( nth @ ( message @ p @ v ) @ ( ft @ ( suc @ index ) ) @ ( minus_minus @ nat @ nMsg @ ( one_one @ nat ) ) ) ) ) ) ).
% AssumptionSubset3(6)
thf(fact_217_EnabledIntermediate,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) @ ( one_one @ nat ) ) @ I4 )
=> ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ I4 ) @ msg ) ) ) ).
% EnabledIntermediate
thf(fact_218_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_219_AssumptionFair_I2_J,axiom,
ord_less @ nat @ n0 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% AssumptionFair(2)
thf(fact_220_Occ4,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) @ nMsg ) @ msg ).
% Occ4
thf(fact_221_AssumptionFair_I3_J,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) @ n0 ) @ msg ).
% AssumptionFair(3)
thf(fact_222_AssumptionSubset3_I4_J,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ nMsg ) @ msg ).
% AssumptionSubset3(4)
thf(fact_223_AssumptionSubset2_I4_J,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ n1 ) @ msgInSet ).
% AssumptionSubset2(4)
thf(fact_224_MessageStaysOrConsumed,axiom,
! [N1: nat,N22: nat,N2: nat,Msg: message @ p @ v] :
( ( ( ord_less_eq @ nat @ N1 @ N22 )
& ( ord_less @ nat @ N22 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ N2 ) ) )
& ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ N2 ) @ N1 ) @ Msg ) )
=> ( ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ N2 ) @ N22 ) @ Msg )
| ? [N03: nat] :
( ( ord_less_eq @ nat @ N1 @ N03 )
& ( ord_less @ nat @ N03 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ N2 ) ) )
& ( ( nth @ ( message @ p @ v ) @ ( ft @ N2 ) @ N03 )
= Msg ) ) ) ) ).
% MessageStaysOrConsumed
thf(fact_225__092_060open_062enabled_A_Ife_A_ISuc_Aindex_J_A_B_A_Ilength_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_J_J_Amsg_092_060close_062,axiom,
enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ ( suc @ index ) ) ) @ ( one_one @ nat ) ) ) @ msg ).
% \<open>enabled (fe (Suc index) ! (length (fe (Suc index)) - 1)) msg\<close>
thf(fact_226_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_227_Occ2,axiom,
enabled @ p @ v @ s @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) ) @ msg ).
% Occ2
thf(fact_228_AssumptionCase1ImplThesis_H,axiom,
enabled @ p @ v @ s @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) ) @ msg ).
% AssumptionCase1ImplThesis'
thf(fact_229_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_230_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_231_length__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_232_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_233_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z2: list @ A] : Y4 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_234_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P2: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X4: A] : ( P2 @ I3 @ X4 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_235_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_236_EnabledOrConsumed,axiom,
( ( enabled @ p @ v @ s @ ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ) @ msg )
| ? [N03: nat] :
( ( ord_less_eq @ nat @ n0 @ N03 )
& ( ord_less @ nat @ N03 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ n ) ) )
& ( ( nth @ ( message @ p @ v ) @ ( ft @ n ) @ N03 )
= msg ) ) ) ).
% EnabledOrConsumed
thf(fact_237_Case2ImplThesis,axiom,
( ? [N0: nat] :
( ( ord_less_eq @ nat @ n0 @ N0 )
& ( ord_less @ nat @ N0 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ n ) ) )
& ( ( nth @ ( message @ p @ v ) @ ( ft @ n ) @ N0 )
= msg ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ n @ N6 )
& ? [N03: nat] :
( ( ord_less_eq @ nat @ n0 @ N03 )
& ( ord_less @ nat @ N03 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ N6 ) ) )
& ( msg
= ( nth @ ( message @ p @ v ) @ ( ft @ N6 ) @ N03 ) ) ) ) ) ).
% Case2ImplThesis
thf(fact_238_EnabledInSuc,axiom,
enabled @ p @ v @ s @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) ) @ msg ).
% EnabledInSuc
thf(fact_239_AssumptionSubset2_I2_J,axiom,
enabled @ p @ v @ s @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) ) @ msgInSet ).
% AssumptionSubset2(2)
thf(fact_240_LastOfIndex,axiom,
( ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ ( suc @ index ) ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ index ) ) ) ).
% LastOfIndex
thf(fact_241_EnabledOrConsumedAtLast,axiom,
( ( enabled @ p @ v @ s @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) ) @ msg )
| ? [N03: nat] :
( ( ord_less_eq @ nat @ n0 @ N03 )
& ( ord_less @ nat @ N03 @ ( size_size @ ( list @ ( message @ p @ v ) ) @ ( ft @ n ) ) )
& ( ( nth @ ( message @ p @ v ) @ ( ft @ n ) @ N03 )
= msg ) ) ) ).
% EnabledOrConsumedAtLast
thf(fact_242_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_243_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_244__092_060open_062_092_060And_062n0_Amsg_O_A_092_060forall_062n_O_Aexecution_OfirstOccurrence_A_Ife_An_J_A_Ift_An_J_Amsg_An0_A_092_060longrightarrow_062_Amsg_A_092_060in_062_D_Amsgs_A_Ilast_A_Ife_An_J_J_092_060close_062,axiom,
! [Msg: message @ p @ v,N04: nat,N7: nat] :
( ( firstOccurrence @ p @ v @ s @ ( fe @ N7 ) @ ( ft @ N7 ) @ Msg @ N04 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( msgs @ p @ v @ s @ product_unit @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ N7 ) ) @ Msg ) ) ) ).
% \<open>\<And>n0 msg. \<forall>n. execution.firstOccurrence (fe n) (ft n) msg n0 \<longrightarrow> msg \<in># msgs (last (fe n))\<close>
thf(fact_245_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_246_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_247_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_248_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_249_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_250__092_060open_062_092_060forall_062n_O_A_092_060forall_062msg_H_092_060in_062firstOccSet_An_O_Amsg_H_A_092_060in_062_D_Amsgs_A_Ilast_A_Ife_An_J_J_092_060close_062,axiom,
! [N7: nat,X5: message @ p @ v] :
( ( member @ ( message @ p @ v ) @ X5 @ ( firstOccSet @ N7 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( msgs @ p @ v @ s @ product_unit @ ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ N7 ) ) @ X5 ) ) ) ).
% \<open>\<forall>n. \<forall>msg'\<in>firstOccSet n. msg' \<in># msgs (last (fe n))\<close>
% Type constructors (16)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A11: $tType] :
( ( order @ A11 )
=> ( order @ ( A10 > A11 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o ).
thf(tcon_List_Olist___Nat_Osize_3,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_4,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_5,axiom,
order @ product_unit ).
thf(tcon_AsynchronousSystem_Omessage___Nat_Osize_6,axiom,
! [A10: $tType,A11: $tType] : ( size @ ( message @ A10 @ A11 ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ nat @ ( minus_minus @ nat @ nMsg @ ( suc @ ( zero_zero @ nat ) ) ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ index ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
%------------------------------------------------------------------------------